Getting rid of Woodward has nothing to do with the man responsible for coaching this team. What you said is true, we have more problems than Jose, doesn't mean we shouldn't sack him. It's not like any external factors are stopping him from coaching his team to play as a team.
Again you mention managers name when the players they played were different, yeah I mean core players, not squad players.
When you have good coach who makes team play as a team, then no one talks about structure and all that BS. These are the excuses when coach is doing a poor job. Liverpool had one good season under Rodgers before going shit, they didn't change much, just hired Klopp. Now everything looks perfect at Liverpool, when before Klopp, Liverpool fans were blaming their owners.
Everything at City remained same, except they signed Pep and they are looking like they will never get beat.
Chelsea also didn't change their structure. They hired Sarri who wanted Rugani, Higuain and ended up with Kovacic on loan, they are playing their best football in years or maybe forever in PL.
Arsenal more or less have same players and have added Emery, they are playing with lot of intensity and there is a clear pattern in their game.
People talk about structure and so many other things, I will never understand how that is related to what happens on the training pitch. Hiring proper DoF won't change how we play. It will only change whom we sign, before turning shit at ManUtd.
Finally, sacking Jose won't fix everything magically but it will be a step in right direction.